In quantum mechanics, the uncertainty principle prevents any non-commuting variables from being described by true joint probability distributions. However, Wigner functions closely approximate these forbidden distributions in several respects. Also, Wigner functions are uniquely related to the quantum mechanical wave function. In other words, Wigner functions can be used to obtain the actual wave function rather than the absolute square of the wave function (commonly known as the probability function).

Transverse spatial Wigner functions of light are “quasi-probability distributions” of two variables which describe photons in a laser beam. The first variable is the photon’s transverse distance from the beam axis. The second variable is the direction in which the photon’s headed. (These variables don’t commute, so a true joint probability distribution is impossible.) The transverse spatial Wigner function of light can be used to obtain the transverse electric field of the beam, rather than its absolute square (i.e. the light intensity).

Every possible beam has a corresponding spatial Wigner function of light. We measured these functions (up to an arbitrary additive constant) using an interferometer built by Dr. Raymer and Dr. Smith. I helped to optimize the design by noticing that if one of the mirrors was removed, the arbitrary constant would change and the output would be inverted but otherwise identical. As a result, the interferometer’s solid angle was increased. I also wrote Maple software that converted the ASCII output of the interferometer’s detector into the surface plots seen in our paper.

In 2003, I gave a PowerPoint presentation about our findings at a conference in Tucson.

Due to time constraints, the presentation didn’t include the following comparisons of theoretical and predicted Wigner functions for a TEM-20 beam:

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Theory

Experiment

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After the conference, I calculated the theoretical spatial Wigner function of a laser beam that had been through a single slit. After modeling the far-field divergence of the beam, my predictions looked like this:

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Theory 1

Theory 2

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Andrew Nahlik and I created this beam in the lab to experimentally test these predictions. Here’s what we found: